# A robust and adaptive MPC formulation for Gaussian process models

arXiv: https://arxiv.org/abs/2507.02098 · HTML: https://mlsanity.com/paper/2507.02098/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 69 references that are in the map.
- First posted on arXiv 2 Jul 2025.

## Builds on (most cited first, 30 of 31)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [acados: a modular open-source framework for fast embedded optimal control](https://mlsanity.com/paper/1910.13753/) (2019)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design](https://mlsanity.com/paper/1503.03144/) (2015)
- [Data-Driven MPC for Quadrotors](https://mlsanity.com/paper/2102.05773/) (2021)
- [Deterministic error bounds for kernel-based learning techniques under bounded noise](https://mlsanity.com/paper/2008.04005/) (2020)
- [ActSafe: Active Exploration with Safety Constraints for Reinforcement Learning](https://mlsanity.com/paper/2410.09486/) (2024)
- [A computationally efficient robust model predictive control framework for uncertain nonlinear systems -- extended version](https://mlsanity.com/paper/1910.12081/) (2019)
- [Automatic nonlinear MPC approximation with closed-loop guarantees](https://mlsanity.com/paper/2312.10199/) (2023)
- [Robust and Kernelized Data-Enabled Predictive Control for Nonlinear Systems](https://mlsanity.com/paper/2206.01866/) (2022)
- [Linear robust adaptive model predictive control: Computational complexity and conservatism -- extended version](https://mlsanity.com/paper/1909.01813/) (2019)
- [Robust adaptive MPC using control contraction metrics](https://mlsanity.com/paper/2209.11713/) (2022)
- [Safe Guaranteed Exploration for Non-linear Systems](https://mlsanity.com/paper/2402.06562/) (2024)
- [Robust Adaptive Model Predictive Control: Performance and Parameter Estimation](https://mlsanity.com/paper/1911.00865/) (2019)
- [A robust adaptive model predictive control framework for nonlinear uncertain systems](https://mlsanity.com/paper/1911.02899/) (2019)
- [State space models vs. multi-step predictors in predictive control: Are state space models complicating safe data-driven designs?](https://mlsanity.com/paper/2203.15471/) (2022)
- [Towards safe and tractable Gaussian process-based MPC: Efficient sampling within a sequential quadratic programming framework](https://mlsanity.com/paper/2409.08616/) (2024)
- [Gaussian processes for dynamics learning in model predictive control](https://mlsanity.com/paper/2502.02310/) (2025)
- [KPC: Learning-Based Model Predictive Control with Deterministic Guarantees](https://mlsanity.com/paper/2011.11303/) (2020)
- [Finite-Sample-Based Reachability for Safe Control with Gaussian Process Dynamics](https://mlsanity.com/paper/2505.07594/) (2025)
- [Zero-Order Optimization for Gaussian Process-based Model Predictive Control](https://mlsanity.com/paper/2211.15522/) (2022)
- [Towards safe Bayesian optimization with Wiener kernel regression](https://mlsanity.com/paper/2411.02253/) (2024)
- [Stochastic Model Predictive Control for Sub-Gaussian Noise](https://mlsanity.com/paper/2503.08795/) (2025)
- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Tube-Certified Trajectory Tracking for Nonlinear Systems With Robust Control Contraction Metrics](https://mlsanity.com/paper/2109.04453/) (2021)
- [L4acados: Learning-based models for acados, applied to Gaussian process-based predictive control](https://mlsanity.com/paper/2411.19258/) (2024)

## Abstract (arXiv)

In this paper, we present a robust and adaptive model predictive control (MPC) framework for uncertain nonlinear systems affected by bounded disturbances and unmodeled nonlinearities. We use Gaussian Processes (GPs) to learn the uncertain dynamics based on noisy measurements, including those collected during system operation. As a key contribution, we derive robust predictions for GP models using contraction metrics, which are incorporated in the MPC formulation. The proposed design guarantees recursive feasibility, robust constraint satisfaction and convergence to a reference state, with high probability. We provide a numerical example of a planar quadrotor subject to difficult-to-model ground effects, which highlights significant improvements achieved through the proposed robust prediction method and through online learning.